Consider the problem of minimizing, over a space of probability measures, the sum of an energy and the entropy, which arises in many situations (models from statistical physics, high-dimensional algorithms...). The associated Wasserstein gradient flow can be interpreted as a nonlinear Langevin process, with the entropy cost leading to Brownian noise. A natural variation of this process with momentum is the underdamped Langevin process, which corresponds to the Vlasov-Fokker-Planck equation. We will see that, for displacement-convex energies, this process achieves a Nesterov acceleration with respect to the gradient flow, meaning that its convergence rate is of the order of the square root of the Polyak-Lojasiewicz constant of the objective function.